On stability of probabilistic automata in environments1

1980 ◽  
Vol 3 (2) ◽  
pp. 117-134
Author(s):  
Jerren Gould
Keyword(s):  
Distribution Function ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Stochastic Matrices ◽  
State Distribution ◽  
Small Perturbations ◽  
Probabilistic Automata ◽  
Cut Point ◽  
The Stability ◽  
Stability Results

Flachs [1], Rabin [7], and Paz [6] have considered topics in the stability of probabilistic automata. Here we extend these results to the more general forms, automata in deterministic environments (ADE). We shall be concerned with two types of stability problems that arise from small perturbations of the environment configurations for an ADE. By consideration of the asymptotic properties of long products of stochastic matrices whose entries are subject to small perturbations concomitant to the environment configuration perturbations, we arrive at sufficient conditions for the state distribution function to be stable (s-stability). In its acceptor formulation the behavior of an ADE is characterized by the sets T ( A, e, λ), the set of tapes accepted by A with environment sequence e and cut-point λ. Sufficient conditions are given for tape-acceptance stability (a-stability) in terms of s-stability and in terms that do not require s-stability. These stability results are pointwise results that give the size of the perturbation of the environment configuration to avoid instability.

ASYMPTOTIC HYPERSTABILITY OF A CLASS OF NEURAL NETWORKS

1999 ◽  
Vol 09 (02) ◽  
pp. 95-98 ◽  
Author(s):  
ANKE MEYER-BÄSE
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lyapunov Methods ◽  
Network Parameters ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Stability Results

This paper is concerned with the asymptotic hyperstability of recurrent neural networks. We derive based on the stability results necessary and sufficient conditions for the network parameters. The results we achieve are more general than those based on Lyapunov methods, since they provide milder constraints on the connection weights than the conventional results and do not suppose symmetry of the weights.

scholarly journals SYNCHRONIZATION OF CHAOTIC FRACTIONAL-ORDER SYSTEMS VIA LINEAR CONTROL

2010 ◽  
Vol 20 (01) ◽  
pp. 81-97 ◽  
Author(s):  
ZAID M. ODIBAT ◽  
NATHALIE CORSON ◽  
M. A. AZIZ-ALAOUI ◽  
CYRILLE BERTELLE
Keyword(s):  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Sufficient Conditions ◽  
Linear Control ◽  
Fractional Order Systems ◽  
Control Laws ◽  
Response System ◽  
The Stability ◽  
Stability Results

The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived analytically to achieve synchronization of the chaotic fractional-order Chen, Rössler and modified Chua systems. Numerical simulations are provided to verify the theoretical analysis.

Zur Stabilität eines Plasmas

1957 ◽  
Vol 12 (10) ◽  
pp. 833-841 ◽  
Author(s):  
K. Hain ◽  
R. Lust ◽  
A. Schlüter
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Sufficient Conditions ◽  
Plasma Cylinder ◽  
Small Perturbations ◽  
Second Order In Time ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Magnetic Field

Die Stabilität von hydrodynamischen Gleichgewichtskonfigurationen wird mit Hilfe der Methode der kleinen Störungen untersucht. Es wird gezeigt, daß das Stabilitätsverhalten durch eine Differentialgleichung 2. Ordnung in der Zeit bestimmt ist, wenn man die Viskosität, den elektrischen Widerstand und die thermische Leitfähigkeit vernachlässigt. Da die Differentialgleichung selbstadjungiert ist, können einige allgemeine Theoreme abgeleitet werden, welche für alle Gleichgewichtskonfigurationen gelten. Man kann zeigen, daß der zeitliche Anstieg von Störungen unter gewissen Bedingungen beschränkt ist. Weiterhin können einige hinreichende Bedingungen für die Stabilität angegeben werden. Für den Spezialfall, daß innerhalb eines Plasmazylinders das Magnetfeld verschwindet, werden die Differentialgleichungen explizit gelöst und Bedingungen für die Stabilität abgeleitet. Schließlich wird auch gezeigt, daß die Differentialgleichung auch selbstadjungiert ist, wenn der Druck nicht isotrop ist.It is shown that the stability of hydromagnetic equilibrium as studied by the method of small perturbations is controlled by one differential equation of second order in time, if one neglects viscosity, electrical resistivity and thermal conductivity. Since the differential equation is self-adjoint some general theorems can be derived which hold for all configurations of hydromagnetic equilibrium. It is possible to show that the rates of growing are limited under certain conditions. Also some sufficient conditions of stability can be given. For a plasma cylinder, inside of which the magnetic field vanishes, the differential equations are solved explicitly and conditions for stability are given. Finally it is shown that the differential equation is also self-adjoint if the pressure is not isotropic.

scholarly journals Stability of a Functional Differential System with a Finite Number of Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Josef Rebenda ◽  
Zdeněk Šmarda
Keyword(s):  
Asymptotic Stability ◽  
Differential System ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Stability Of Solutions ◽  
Functional Differential ◽  
The Stability ◽  
Complex Valued ◽  
Functional Differential System

The paper is devoted to the study of asymptotic properties of a real two-dimensional differential system with unbounded nonconstant delays. The sufficient conditions for the stability and asymptotic stability of solutions are given. Used methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied.

New results for global stability of neutral-type delayed neural networks

2018 ◽  
Author(s):  
Neyir Ozcan
Keyword(s):  
Neural Networks ◽  
Time Delays ◽  
Matrix Theory ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Neural System ◽  
Delayed Neural Networks ◽  
The Matrix ◽  
The Stability ◽  
Stability Results

"This paper deals with the stability analysis of the class of neutral-type neural networks with constant time delay. By using a suitable Lyapunov functional, some delay independent sufficient conditions are derived, which ensure the global asymptotic stability of the equilibrium point for this this class of neutral-type neural networks with time delays with respect to the Lipschitz activation functions. The presented stability results rely on checking some certain properties of matrices. Therefore, it is easy to verify the validation of the constraint conditions on the network parameters of neural system by simply using some basic information of the matrix theory."

scholarly journals New results for a coupled system of ABR fractional differential equations with sub-strip boundary conditions

2021 ◽  
Vol 7 (3) ◽  
pp. 4386-4404
Author(s):  
Mohammed A. Almalahi ◽  
◽  
Satish K. Panchal ◽  
Tariq A. Aljaaidi ◽  
Fahd Jarad ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Uniqueness Of Solutions ◽  
Alternative Theorem ◽  
Fractional Differential ◽  
Mathematical Techniques ◽  
The Stability ◽  
Stability Results ◽  
Banach’S Contraction Principle

<abstract><p>In this article, we investigate sufficient conditions for the existence, uniqueness and Ulam-Hyers (UH) stability of solutions to a new system of nonlinear ABR fractional derivative of order $ 1 &lt; \varrho\leq 2 $ subjected to multi-point sub-strip boundary conditions. We discuss the existence and uniqueness of solutions with the assistance of Leray-Schauder alternative theorem and Banach's contraction principle. In addition, by using some mathematical techniques, we examine the stability results of Ulam-Hyers (UH). Finally, we provide one example in order to show the validity of our results.</p></abstract>

scholarly journals New Exponential Stability Results for Neutral Stochastic Systems with Distributed Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Jian Wang
Keyword(s):  
Exponential Stability ◽  
Stability Problem ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Distributed Delays ◽  
Grey System ◽  
Mean Square Exponential Stability ◽  
Neutral Stochastic Systems ◽  
The Stability ◽  
Stability Results

This paper studies the stability problem of a grey neutral stochastic system with distributed delays. The main feature of a grey system is that the parameters of system are evaluated by grey numbers, and system has the strong background in engineering applications. However, the literature dealing with the stability problem for grey system seems to be scarce. The main aim of this paper is to fill the gap. Based on the Lyapunov stability theorem and Itô’s formula, especially, using the decomposition technique of the continuous matrix-covered sets of grey matrix, we propose several novel sufficient conditions, which ensure our considered grey system in mean-square exponential stability and almost surely exponential stability. Furthermore, two examples are provided to show the effectiveness of the obtained results.

Stability and Regularization of Difference Schemes in Banach and Hilbert Spaces

2013 ◽  
Vol 13 (2) ◽  
pp. 139-160
Author(s):  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov
Keyword(s):  
Banach Space ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Iteration Scheme ◽  
Difference Schemes ◽  
Stability Conditions ◽  
Initial Operator ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Stability Results

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

Epitaxial Growth in Dislocation-Free Strained Alloy Films

2002 ◽  
Vol 715 ◽  
Author(s):  
Zhi-Feng Huang ◽  
Rashmi C. Desai
Keyword(s):  
Deposition Rate ◽  
Elastic Moduli ◽  
Composition Dependence ◽  
Critical Thickness ◽  
Lattice Misfit ◽  
Misfit Strain ◽  
Joint Stability ◽  
Alloy Films ◽  
The Stability ◽  
Stability Results

AbstractThe morphological and compositional instabilities in the heteroepitaxial strained alloy films have attracted intense interest from both experimentalists and theorists. To understand the mechanisms and properties for the generation of instabilities, we have developed a nonequilibrium, continuum model for the dislocation-free and coherent film systems. The early evolution processes of surface pro.les for both growing and postdeposition (non-growing) thin alloy films are studied through a linear stability analysis. We consider the coupling between top surface of the film and the underlying bulk, as well as the combination and interplay of different elastic effects. These e.ects are caused by filmsubstrate lattice misfit, composition dependence of film lattice constant (compositional stress), and composition dependence of both Young's and shear elastic moduli. The interplay of these factors as well as the growth temperature and deposition rate leads to rich and complicated stability results. For both the growing.lm and non-growing alloy free surface, we determine the stability conditions and diagrams for the system. These show the joint stability or instability for film morphology and compositional pro.les, as well as the asymmetry between tensile and compressive layers. The kinetic critical thickness for the onset of instability during.lm growth is also calculated, and its scaling behavior with respect to misfit strain and deposition rate determined. Our results have implications for real alloy growth systems such as SiGe and InGaAs, which agree with qualitative trends seen in recent experimental observations.

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